Asymptotic behavior of the solution of the fractional heat equation

Abstract

In this short note, we study the asymptotic behavior of the solution to the fractional heat equation ∂u∂t(t,x)=−(−Δ)α/2u(t,x)+u(t,x)∂d+1W∂t∂x1⋯∂xd,t>0,x∈Rd with initial condition bounded above and below, where α∈(0,2) and the noise W(t,x) is a fractional Brownian sheet. The method of moments is used to investigate the solution.